No checks are necessary unless the wildshaped druid is trying to conceal an action that is decisively unnnatural to the wildshaped form.

There is some guidance in the Dungeon Master's Guide section Using Ability Scores:

When a player wants to do something, it’s often appropriate to let the attempt succeed without a roll or a reference to the character’s ability scores. For example, a character doesn’t normally need to make a Dexterity check to walk across an empty room or a Charisma check to order a mug of ale. Only call for a roll if there is a meaningful consequence for failure.

When deciding whether to use a roll, ask yourself two questions:

• Is a task so easy and so free of conflict and stress that there should be no chance of failure?
• Is a task so inappropriate or impossible — such as hitting the moon with an arrow — that it can’t work?

If the answer to both of these questions is no, some kind of roll is appropriate.

Say you wildshape into a goat. You have to ask yourself, "does my enemy have any reason to believe that this goat that is physically indistinguishable from a goat is not really a goat? Remember, your physical characteristics are exactly that of a goat, therefore unless you say you are doing something non-goat-like, there is a reasonable expectation that you are moving and acting as a goat would. Think about it this way: there's obviously no check required to see if you know how to use this totally unique musculo-skeletal system you now possess to walk around. The ability to move and use your body as a goat naturally would is part-and-parcel to Wildshape.

Therefore, the answer to the first question in the DMG guidance would be "yes, a goat acting like a goat is so easy and so free of stress there should be no chance of failure." It is when a goat starts acting like a goat would not that checks need to be made: How well can I make this non-goat-like action appear to be goat-like?

This is an interesting situation. Although you can use the 'normal' checks as listed above, I would take a slightly different approach (although still able to be adjudicated through the normal rules). Sometimes the basic attributes for a skill don't cover all its uses.

Which Skill?

I would base the skill proficiency on the type of animal you are trying to emulate. If you are trying to emulate a domesticated animal that is possibly trained in tricks, Animal Handling seems the most appropriate skill to use. If you are trying to emulate an animal that is wild and trying to do so without appearing to have more intellect than it does then Nature may be a better option. If you are trying to emulate something truly extraordinary, such as an elemental or other magical creature (polymorph), then Arcana is likely the best skill option. Choose the skill that is most likely used to identify that creature and its actions and you should be on the right track.

Fooling The Enemy

Any time you are trying to trick your enemy into thinking what you want, you need Charisma, so this choice is obvious. Yes, it is a form of deception, so that skill may be appropriate in some ways, but I would say a druid is likely to be far better served by knowing how the animal acts and acting based on that.

Due to this my suggestion is a Charisma check with proficiency modifier based on the "Which Skill?" decision above.

Detection:

When observing your surroundings, Wisdom is the go to stat, so this is the right stat to look at here (unless suspicion is aroused, then Intelligence may be a better option such as an investigation check would lend). Accompany this with an appropriate skill as noted previously and you should be on the right track.

Wisdom check with proficiency modifier based on appropriate skill.

Other possible methods:

Another possibility is giving advantage to 'normal' skill checks if they are proficient in a related skill, but this seems a bit over the top to me. You end up giving out some form of advantage/disadvantage on every single roll.

Finally, you can just use standard skill checks as they are normally rolled. I find it a bit odd to use an Intelligence(Nature) check to determine if someone is an animal, but a Wisdom(Insight) check doesn't quite seem appropriate. Charisma(Deception) seems wildly inappropriate for a druid who wants to act like a creature they're likely very intimate with the actions of in the first place, so I lean towards my above suggestions.

Wrap it up (TL:DR)

Summation (uses a horse or dog in examples):

1. Choose a skill appropriate for knowledge about the creature and its behaviors.

   • (Animal Handling)

2. Shape-Shifter rolls Charisma check, with proficiency as noted above. This is the DC of the observer's check.

   • (Charisma(Animal Handling) check)

3. Observer gets a passive check if not actively searching.

   • (10 + Wisdom modifier + (Animal Handling))

   Observer rolls actively if trying to detect abnormalities.

   • (Wisdom(Animal Handling) check)

Caveat

This is only to cover how to deal with the roll. My general adjudication would say that if they are significantly more proficient, and have advantage (due to role-playing for example), then there is no need for a roll.

I have no hard and fast rule on that, but an obvious scenario would be the trained and specialized druid (+3 Charisma, +2 prof bonus) with advantage (+5 to passives) would have an "average roll" of 20. That is a no-brainer in situations where the bonus and advantage give 10 points more for the shape-shifter than the observer has. A close check is a bit different, however. An average guard would have an average roll of 11-13. This means that they would need about a 17 or higher to notice the difference. Let them default the effect with no roll.

It's not really that hard, especially in 5th edition, to keep track of these numbers. In this case it's Charisma mod, proficiency, and advantage/disadvantage. If all are positive, lean towards defaulting the roll to success.

I would predicate the rolls based on what the shape changed character is doing and whether or not the observers would be expected to know anything about that animal. A cat lazily sitting by the road near a village ought to be almost impossible for the druid to screw up, but a deer wandering through a camp during the day, that might provoke some rolls (or at least an improptu hunting attempt!).

Does the observer have any particular experience with the animal? The local kennelmaster might have a much better chance to detect the druid as a dog (or at least think something is wrong with it, like rabies [kill it with fire!]) than an out of town merchant with no particular interest in dogs.

WRT the ambush scenario, are the ambushees planning to attack the town? In that case they may want to silence any dogs, strange behavior or not. Are they from the town? They might wonder about a strange dog showing up. In the wilderness, a dog implies people nearby, so that might alert the ambushees more than something odd about the dog itself.

A contested deception vs nature/insight roll seems like a good way to resolve any uncertainty, with perhaps advantage/disadvantage given to represent specific familiarity or unfamiliarity with the animal/area in question. But even pretty obtrusive actions by the character, if paired with an appropriate animal (like a raccoon rooting through an enemies stuff) shouldn't necessarily provoke a roll, but should generate a reaction.

A second consideration is whether or not the wild shape is detectable as magic. Certainly any worn magic items should be detectable, but is the use of wild shape itself a lingering magic effect that can be detected? The description states it is a magic effect and Detect Magic says "creature...that bears magic", so I'd say it would be detectable.

Tl;dr - The Druid makes a Charisma (deception) check, an observer has to beat their result with a Wisdom (insight) check. Proficiency in Nature would allow a character who was not proficient to add their proficiency bonus to the whichever of the 2 checks they were making.

So, there are 2 questions here. The first is whether the Wild Shaped Druid is physically indistinguishable from a normal animal. Wild Shape says that you assume the form of a regular animal, not a Dire, celestial, fey, or anything else. So I think we can take it as read that if you Wild Shape into a dog, you look like a dog, because, well, on a physical level, you are a dog.

The second, more interesting question, is the behavioural issue. Can an intelligent humanoid act exactly like a regular animal? Given that you retain your mental scores and the shape you assume doesn't influence your actions in any way, it seems clear that physically being a dog doesn't make you automatically act like one. So our Druid has to pretend to be a dog. This works exactly like Alter Self, which lets you make yourself appear as a member of another race. If you used Alter Self to pretend to be the local ruler, it would be assumed by all that Charisma (deception) checks would be required to be convincing. Note that Alter Self doesn't give any benefit to this check - it just fixes your appearance for you.

Now, acting is covered by Charisma (Deception) checks. So the obvious call would be to have the Druid make a Charisma (deception) check, which any observer would have to make a Wisdom (Insight) check to see through. This would probably work, but there is also Intelligence (Nature) to consider. Does the observer know how an animal of that type normally behaves?

There are a few options here. The first is to say that the observer has to make both an Intelligence (nature) check and a Wisdom (insight) check. The second is that the observer either has to make a Wisdom (insight) check or an Intelligence (nature) check. I would say that knowing how an animal normally behaves isn't the same as noticing that it is behaving oddly. Therefore, if I was going to include an Intelligence (nature) check, I would require it in addition to a Wisdom (insight) check.

On the other hand, I don't know much about dogs, but I could probably notice if one was, e.g., carefully watching everyone who went past. So I would say that Wisdom (insight) is all that is required. Perhaps an Intelligence (nature) check could be used to gain advantage on the Wisdom (insight) check, or proficiency in Nature could allow a character not proficient in Insight to add their proficiency bonus to the Wisdom roll. That last is what I'd rule.

Whatever decision you use for Nature to influence the Wisdom check, for the sake of both fairness and consistency the Druid should have the same benefit to their Charisma check. So I would allow a Druid proficient in Nature to add their proficiency bonus to the Charisma check even if they weren't proficient in Deception.